Nonoscillation theory of elliptic equations of order 2n

Walter Allegretto
1976 Pacific Journal of Mathematics  
Several nonoscillation theorems are obtained for elliptic equations of order In. These results extend several well known nonoscillation theorems for elliptic equations of order 2 and 4, and for ordinary differential equations of higher order. Introduction. Several authors have considered the problem of establishing oscillation and nonoscillation criteria for elliptic equations. We refer the reader to the books by C. A. Swanson [15] and K. Kreith [8] where extensive bibliographies can be found.
more » ... ost of the interest has so far centered on second order equations, with some results also established for fourth order equations. In this paper we establish several nonoscillation theorems for elliptic equations of order In. These theorems extend in particular, results of Swanson [14], Piepenbrink [12], Headley and Swanson [5] and Yoshida [16]. Our proofs make extensive use of variational arguments, of extended Sobolev-type inequalities and of estimates on quadratic forms associated with elliptic equations. The first part of the paper discusses some preliminary comparison theorems and lower estimates on quadratic forms. The second part deals with the nonoscillation of operations defined in subdorcteins of E m for m^2. In the next part, some results are established for operations defined in subdomains of E 2 . The final part deals with extensions to more general cases. Definitions and notations. Let Ω be an unbounded domain Without loss of generality, we may assume 0 g: Ω. Points of E m are denoted by x = (jc b , x m ) and differentiation with respect to JC, by D t , ΐ = 1, '-,m. Let L be the differential expression given by:
doi:10.2140/pjm.1976.64.1 fatcat:6a7qq76uhrhtlbm25mwsqtzkfe